416 research outputs found
Irregular conformal blocks, with an application to the fifth and fourth Painlev\'e equations
We develop the theory of irregular conformal blocks of the Virasoro algebra.
In previous studies, expansions of irregular conformal blocks at regular
singular points were obtained as degeneration limits of regular conformal
blocks; however, such expansions at irregular singular points were not clearly
understood. This is because precise definitions of irregular vertex operators
had not been provided previously. In this paper, we present precise definitions
of irregular vertex operators of two types and we prove that one of our vertex
operators exists uniquely. Then, we define irregular conformal blocks with at
most two irregular singular points as expectation values of given irregular
vertex operators. Our definitions provide an understanding of expansions of
irregular conformal blocks and enable us to obtain expansions at irregular
singular points.
As an application, we propose conjectural formulas of series expansions of
the tau functions of the fifth and fourth Painlev\'e equations, using
expansions of irregular conformal blocks at an irregular singular point.Comment: 26 page
Bilinear Discrete Painleve-II and its Particular Solutions
By analogy to the continuous Painlev\'e II equation, we present particular
solutions of the discrete Painlev\'e II (d-P) equation. These
solutions are of rational and special function (Airy) type. Our analysis is
based on the bilinear formalism that allows us to obtain the function
for d-P. Two different forms of bilinear d-P are obtained
and we show that they can be related by a simple gauge transformation.Comment: 9 pages in plain Te
Integrable nonlinear equations on a circle
The concept of integrable boundary value problems for soliton equations on
and is extended to bounded regions enclosed by
smooth curves. Classes of integrable boundary conditions on a circle for the
Toda lattice and its reductions are found.Comment: 23 page
Complex sine-Gordon-2: a new algorithm for multivortex solutions on the plane
We present a new vorticity-raising transformation for the second integrable
complexification of the sine-Gordon equation on the plane. The new
transformation is a product of four Schlesinger maps of the Painlev\'{e}-V to
itself, and allows a more efficient construction of the -vortex solution
than the previously reported transformation comprising a product of maps.Comment: Part of a talk given at a conference on "Nonlinear Physics. Theory
and Experiment", Gallipoli (Lecce), June-July 2004. To appear in a topical
issue of "Theoretical and Mathematical Physics". 7 pages, 1 figur
On the extended resolvent of the Nonstationary Schrodingher operator for a Darboux transformed potential
In the framework of the resolvent approach it is introduced a so called
twisting operator that is able, at the same time, to superimpose \`a la Darboux
solitons to a generic smooth decaying potential of the Nonstationary
Schr\"odinger operator and to generate the corresponding Jost solutions. This
twisting operator is also used to construct an explicit bilinear representation
in terms of the Jost solutions of the related extended resolvent. The main
properties of the Jost and auxiliary Jost solutions and of the resolvent are
discussed.Comment: 24 pages, class files from IO
Linearizability of the Perturbed Burgers Equation
We show in this letter that the perturbed Burgers equation is equivalent, through a near-identity transformation and
up to order \epsilon, to a linearizable equation if the condition is satisfied. In the case this
condition is not fulfilled, a normal form for the equation under consideration
is given. Then, to illustrate our results, we make a linearizability analysis
of the equations governing the dynamics of a one-dimensional gas.Comment: 10 pages, RevTeX, no figure
Ablowitz-Ladik system with discrete potential. I. Extended resolvent
Ablowitz-Ladik linear system with range of potential equal to {0,1} is
considered. The extended resolvent operator of this system is constructed and
the singularities of this operator are analyzed in detail.Comment: To be published in Theor. Math. Phy
A Bi-Hamiltonian Formulation for Triangular Systems by Perturbations
A bi-Hamiltonian formulation is proposed for triangular systems resulted by
perturbations around solutions, from which infinitely many symmetries and
conserved functionals of triangular systems can be explicitly constructed,
provided that one operator of the Hamiltonian pair is invertible. Through our
formulation, four examples of triangular systems are exhibited, which also show
that bi-Hamiltonian systems in both lower dimensions and higher dimensions are
many and varied. Two of four examples give local 2+1 dimensional bi-Hamiltonian
systems and illustrate that multi-scale perturbations can lead to
higher-dimensional bi-Hamiltonian systems.Comment: 16 pages, to appear in J. Math. Phy
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